Even though the Standard Model (SM) is weakly coupled at the Fermi scale, a new strong dynamics involving its degrees of freedom may conceivably lurk at slightly higher energies, in the multi TeV range. Approximate symmetries provide a structurally robust context where, within the low energy description, the dimensionless SM couplings are weak, while the new strong dynamics manifests itself exclusively through higher-derivative interactions. We present an exhaustive classification of such scenarios in the form of effective field theories, paying special attention to new classes of models where the strong dynamics involves, along with the Higgs boson, the SM gauge bosons and/or the fermions. The IR softness of the new dynamics suppresses its effects at LEP energies, but deviations are in principle detectable at the LHC, even at energies below the threshold for production of new states. Our construction provides the so far unique structurally robust context where to motivate several searches in Higgs physics, diboson production, or WW scattering, which were so far poorly justified. Perhaps surprisingly, the interplay between weak coupling, strong coupling and derivatives, which is controlled by symmetries, can override the naive expansion in operator dimension, providing instances where dimension-8 dominates dimension-6, well within the domain of validity of the low energy effective theory. This result reveals the limitations of an analysis that is both ambitiously general and restricted to dimension-6 operators.